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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$.\n$E$ is an *F-space* if there is a [[metric|063W5JMY85]] $d$ on $E$ such that:\n\n1. $d$ induces the topology of $E$.\n2. $d$ is translation-invariant: $d(x + z, y + z) = d(x, y)$ for all $x, y, z \\in E$.\n3. The metric space $(E, d)$ is [[complete|08EWT7J0SC]].\n\n## Remarks\n\nSome authors use the name for a vector space with a metric satisfying 2 and 3 for which addition and scalar multiplication are continuous; this is the same notion.\nExample: every Banach space is an F-space with $d(x, y) = \\norm{x - y}$; this metric induces the norm topology, it is translation-invariant since $(x + z) - (y + z) = x - y$, and it is complete by definition of a Banach space.\nExample: for $0 \u003c p \u003c 1$, $L^p([0, 1])$ with $d(f, g) = \\int_0^1 \\abs{f - g}^p$ is an F-space that is not locally convex.\nNon-example: the space $c_{00}$ of finitely supported real sequences with $d(x, y) = \\sup_n \\abs{x_n - y_n}$ satisfies 1 and 2 but not 3: the sequences $x^{(k)} = (1, 1/2, \\ldots, 1/k, 0, 0, \\ldots)$ are Cauchy and have no limit in $c_{00}$.\nEvery metrizable topological vector space has a translation-invariant metric inducing its topology, so condition 2 matters only together with condition 3.\n",
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